Solving Systems of Linear Equations: Substitution & Elimination

Linear Equations

A set of equations with the same variables, typically two equations in two variables.

Substitution Method

A technique for solving linear equations where one equation is solved for one variable and substituted into the other equation.

Elimination Method

A method for solving systems of equations by combining the equations to eliminate one variable.

Standard Form

The form of a linear equation written as Ax + By = C, where A, B, and C are constants.

Isolate a Variable

To rearrange an equation so that one variable is alone on one side of the equation.

Opposite Coefficients

When the coefficients of a variable in two equations are equal in magnitude but opposite in sign.

Back-Substitution

The process of substituting a known value back into an equation to find another variable.

Check Your Work

The process of verifying the solution by substituting values back into the original equations.

Population Equation

An equation that represents the population as a function of time, such as $P = 14t^2 + 820t + 42000$.

Quadratic Formula

The solution method for quadratic equations given by $t = \frac{-b \boldsymbol{ ext{±}} ext{sqrt}(b^2 - 4ac)}{2a}$.

Discriminant

The part of the quadratic formula that determines the nature of the roots, given by $D = b^2 - 4ac$.

Variable

A symbol (often a letter) that represents an unknown value in mathematical expressions.

Equation

A mathematical statement that asserts the equality of two expressions.

Slope-Intercept Form

The form of a linear equation written as $y = mx + b$ where m is the slope and b is the y-intercept.

Coefficient

A numerical or constant quantity placed before and multiplying a variable in an equation.

Linear Equation

An equation that makes a straight line when graphed and has no variables raised to a power greater than one.

Step-by-Step Instructions

A series of ordered actions to guide through a process or method.

System Solution

The pair of values (x, y) that satisfies all equations in a system of linear equations.

Discriminant Value

A value derived from the coefficients of a quadratic equation that shows if the roots are real, complex, or repeated.

Time Variable

A variable that represents time in relation to another parameter, often denoted as t.

Substitution Example

An example showing how one can find a variable by substituting known values into equations.

Solving for Remaining Variable

The process of manipulating an equation to find the value of the variable that has not yet been determined.

Identify the Problem

The first step in solving a problem, focusing on its key elements and requirements.

Algebraic Manipulation

The process of rearranging and simplifying equations to solve for variables.

Multiple Solutions

When more than one set of values satisfies the equations in a system.

Final Answer Presentation

The last step where the solution is expressed clearly, including any required units.

Population Analysis

The study of population trends using mathematical models, like equations.

Calculation of Roots

The process of finding the values for a variable by applying the quadratic formula.

Positive Assessment

Confirmation that a solution is valid and meets the criteria provided in a word problem.

Linear system simplification

The act of reducing a system of equations into a simpler form to make solving easier.

Distinct Solutions

Unique solution(s) distinct from other potential answers in systems of equations.

Graphing Equations

Visualizing equations on a coordinate plane to find intersection points, representing solutions.

Variables and Relationships

Understanding how different quantities are connected within equations and their respective contexts.

Equation Rearrangement

The process of manipulating an equation to isolate a specific variable or set of terms.

Solution Verification

The act of checking whether a derived solution satisfies the conditions outlined in the original equations.

Analysis of Results

The process of reviewing outcomes to derive conclusions and understand implications.

Stepwise Consideration

 Attention to each step in a methodical fashion to ensure comprehensive understanding and correctness.